Timeline for pullback of a local system
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Dec 9, 2019 at 14:20 | comment | added | Will Sawin | @AnhDũngLê Exactly. | |
| Dec 9, 2019 at 13:40 | comment | added | Anh Dũng Lê | also since I am working with smooth projective $k$-scheme and smooth over a field implies normal, thus implies geometrically unibranch. We can use lemma 7.4.10, whose proof basically says that any geometric cover $Z$ of $X$ is a disjoint union of finite étale covers. Since the pullback of $F$ on $Z$ is trivial, the pullback of $F$ on each disjoint component is trivial too. Is that correct? | |
| Dec 9, 2019 at 13:26 | comment | added | Will Sawin | @AnhDũngLê Yeah, that's what I'm suggesting. The point is because it is continuous on a discrete set, the stabilizer is open, and because it has a finite basis, the kernel is a finite intersection of stabilizers and is open. | |
| Dec 9, 2019 at 13:08 | comment | added | Anh Dũng Lê | Can you please check if I understand this approach correctly? I have a local system $F$ on $X_{\acute{e}t}$ with values in a $k$-vector space $V$, also a local system in $X_{pro\acute{e}t}$. This correponds to a geometric cover $Y$, which is an étale $X$-scheme. The fiber of each point is $V$ as a $\pi_1^{proet}(X,x)$-set and the action of the group gives rise to a representation. Let $U$ be the kernel of the representation, then $\pi_1^{proet}(X,x)/U$ is also a $\pi_1^{proet}(X,x)$-set. This set corresponds to another geometric cover $Z$ and the pullback of $F$ on $Z$ is trivial. | |
| Dec 8, 2019 at 1:38 | comment | added | Will Sawin | @AnhDũngLê Lemma 7.4.5 there works for sheaves of sets, which includes arbitrary representations. The local field lemma is needed because the topology on the local field may be relevant. But the topology on $\mathbb C$ is never relevant. | |
| Dec 8, 2019 at 0:33 | comment | added | Anh Dũng Lê | Thank you very much, lemma 7.4.7. only works for local fields. Do we know anything about other like $\mathbb{C}$? | |
| Dec 7, 2019 at 23:02 | comment | added | Will Sawin | @AnhDũngLê This is probably extreme overkill but you can get this out of Lemmas 7.4.5 and 7.4.10 of arxiv.org/pdf/1309.1198.pdf | |
| Dec 7, 2019 at 22:51 | comment | added | Will Sawin | @AnhDũngLê Oh I see what you mean - sorry I misinterpreted. I think it should work as long as you take continuous representations with the discrete topology on $k$. But I don't know why would want to do this.. | |
| Dec 7, 2019 at 20:24 | comment | added | Anh Dũng Lê | but proposition 6.15 is for topological spaces and the next one 6.16 is for connected scheme, but for locally constant sheaf of finite stalks, so if I work with k-vector spaces of characteristic 0, then it is infinite. | |
| Dec 7, 2019 at 19:45 | comment | added | Will Sawin | @AnhDũngLê In the online version jmilne.org/math/CourseNotes/LEC.pdf Proposition 6.15 - it of course works over an arbitrary base field. | |
| Dec 7, 2019 at 18:17 | comment | added | Anh Dũng Lê | Can you specify where in Milne's étale cohomology? I couldn't find it. Does $k$ have to be a finite field for this to work or does it also work for $k$ of characteristic 0. | |
| Dec 5, 2019 at 20:57 | vote | accept | Anh Dũng Lê | ||
| Dec 5, 2019 at 15:48 | history | answered | Will Sawin | CC BY-SA 4.0 |